First Borel class sets in Banach spaces and the asymptotic-norming property

Abstract

The Radon-Nikodým property in a separable Banach spaceX is related to the representation ofX as a weak* first Borel class subset of some dual Banach space (its bidualX**, for instance) by well known results due to Edgar and Wheeler [8], and Ghoussoub and Maurey [9, 10, 11]. The generalizations of those results depend on a new notion of Borel set of the first class “generated by convex sets” which is more suitable to deal with non-separable Banach spaces. The asymptotic-norming property, introduced by James and Ho [13], and the approximation by differences of convex continuous functions are also studied in this context.

P. W. McCartney and R. C. O’Brien,A separable Banach space with the Radon-Nikodým property that is not isomorphic to a subspace of a separable dual, Proceedings of the American Mathematical Society78 (1980), 40–42.MATHCrossRefMathSciNetGoogle Scholar